# Unsupervised Learning in RSS-Based DFLT Using an EM Algorithm

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## Abstract

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## 1. Introduction

- it is passive and does not require users to carry obtrusive and uncomfortable sensors [2];
- RF signals can penetrate walls, clutter and other occlusions, unlike many other sensors that have a limited field of view [8]; and
- it is privacy preserving which increases acceptance of the monitoring technology, unlike vision-based systems that are intrusive [9].

- A Gaussian filter is presented to estimate the state of the target and a novel Measurement Selection unit is developed to select and combine the measurement models of two DFLT methods into one filtering algorithm. The developed system is demonstrated to outperform a state-of-the-art adaptive DFLT system and reduce the tracking error by $42\%$.
- A Gaussian smoother is implemented, and it is used to evaluate the expectations involved in the expectation step of the Expectation-Maximization (EM) algorithm. Moreover, we show how the maximization step of the EM algorithm is available in closed form for the considered measurement model. The presented EM algorithm is computationally very efficient, up to 18 times faster than current solutions used in the literature.
- An EM algorithm is presented for estimating the unknown RSS model parameters, liberating the system from the need for supervised training and calibration periods. It is demonstrated that the EM algorithm not only improves the accuracy of the introduced system, but also other DFLT systems.
- The experiments conducted in this paper, together with Matlab code to run the presented filtering, smoothing and EM algorithms are made publicly available and are published in [21]. The aim is to lower the threshold to start research in the area and advance the field of DFLT in general.

## 2. Related Work

## 3. Localization and Tracking

#### 3.1. Models

#### 3.1.1. Dynamic Model

#### 3.1.2. Measurement Model

#### 3.2. Radio Tomographic Imaging

#### 3.2.1. Image Estimation

#### 3.2.2. RTI Positioning

#### 3.3. Tracking Filter

#### 3.3.1. EKF Observation Model

#### 3.3.2. Prediction Step

#### 3.3.3. Measurement Update

#### 3.4. Measurement Selection

**if**${\u03f5}_{1}>\mathcal{T}$ and ${\u03f5}_{2}\le \mathcal{T}$—It is likely that the filter has diverged. Use only the output of RTI, i.e., $\mathsf{R}={\mathbf{C}}_{k}$, $\mathsf{H}=\mathbf{H}$ and ${\mathit{\nu}}_{k}={\widehat{\mathbf{p}}}_{k}-\mathbf{H}{\mathbf{m}}_{k}^{-}$.**else if**${\u03f5}_{1}\le \mathcal{T}$—Normal operation, concatenate the models: $\mathsf{R}=\mathrm{blkdiag}\left(\right)open="("\; close=")">{\mathbf{C}}_{k},\mathbf{R}$, $\mathsf{H}={\left[\begin{array}{cc}{\mathbf{H}}^{T}& {\mathbf{H}}_{\mathbf{x}}^{T}\end{array}\right]}^{T}$ and ${\mathit{\nu}}_{k}={\left[\begin{array}{c}{\left(\right)}^{{\widehat{\mathbf{p}}}_{k}}T\\ {\left(\right)}^{{\mathbf{y}}_{k}}T\end{array}\right]}^{}$**else**—The RTI position estimate is likely inaccurate, use only the RSS measurements, i.e., $\mathsf{R}=\mathbf{R}$, $\mathsf{H}={\mathbf{H}}_{\mathbf{x}}$ and ${\mathit{\nu}}_{k}={\mathbf{y}}_{k}-\mathcal{H}\left({\mathbf{m}}_{k}^{-}\right)\mathit{\theta}$.

## 4. Parameter Estimation

#### 4.1. Gaussian Smoothing

#### 4.2. Expectation-Maximization-Based Parameter Estimation

## 5. Experiments

## 6. Experimental Results

#### 6.1. EM with Existing DFLT Methods

#### 6.2. Parameter Estimation Algorithms

#### 6.3. System Comparison

#### 6.4. System Performance over Time

#### 6.5. Simulations

## 7. Conclusions

## Appendix A. EM Algorithm

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Measured ( ) and modeled ( ) RSS as a function of excess path length for two example links. The shaded region depicts the $3\sigma $ confidence interval. In (

**a**), the model parameters are $\mathbf{\Theta}=[-67.02,3.42,0.03,3.92]$ and in (

**b**), $\mathbf{\Theta}=[-56.61,-5.71,0.04,0.80]$.

**Figure 4.**Two example RTI images and the position and covariance estimates calculated using (12) and (13). In the image, the deep blue regions indicate areas that are not occupied by people and the bright regions indicate estimated obstructions. Furthermore, the plus sign indicates the true position, the crosses are the position estimates, and the dashed line illustrates the $3\sigma $ uncertainty ellipse [3], reprinted with permission from ref. [3]. Copyright 2019 IEEE.

**Figure 6.**The RMSE, averaged over the 18 different experiments, as a function of EM iteration number. The DFLT methods are: EKF ( ), RTI ( ) and PF ( ).

**Figure 7.**Comparison of different parameter estimation algorithms as a function of iteration number: EM(${\mathbf{\Theta}}^{\left\{1\right\}}$) ( ), NLS(${\mathbf{\Theta}}^{\left\{1\right\}}$) ( ), NLS(${\mathbf{\Theta}}^{\left\{2\right\}}$) ( ) and NLS(${\mathbf{\Theta}}^{\left\{3\right\}}$) ( ).

**Figure 8.**Tracking accuracy of the system in Ex$6.3$. In the figure, the ground truth coordinates are shown using ( ), the estimated with ( ) and the pink area illustrates the $3\sigma $ confidence interval of RTI position estimates.

**Figure 9.**The posterior Cramér-Rao bound ( ) and RMSE of the proposed system ( ) and the benchmark system ( ). For every iteration number, the results are averaged over 100 Monte Carlo simulations.

Parameter | Unit | |
---|---|---|

Pixel Variance (9) | ${\sigma}_{b}^{2}$ | $0.0005$ (dB${}^{2}$) |

Correlation distance (9) | ${\delta}_{d}$ | $0.5$ (m) |

Spatial decay rate (10) | $\lambda $ | $0.04$ (m) |

Pixel width (7) | ${\delta}_{p}$ | $0.25$ (m) |

Image threshold (12) | $\gamma $ | $0.70$ |

EKF | ARTI | |
---|---|---|

Ex1 (open) | $31.3\pm 18.7$ | $62.8\pm 46.8$ |

Ex2 (open) | $20.8\pm 11.4$ | $26.0\pm 15.0$ |

Ex3 (open) | $17.2\pm 8.9$ | $23.8\pm 11.5$ |

Ex4 (apt.) | $50.4\pm 28.0$ | $85.1\pm 62.0$ |

Ex5 (apt.) | $40.1\pm 20.9$ | $62.9\pm 41.3$ |

Ex6 (apt.) | $36.7\pm 21.1$ | $49.6\pm 28.4$ |

Trial 1 | Trial 2 | Trial 3 | Average | |
---|---|---|---|---|

Ex1 (open) | $23.4$ | $25.1$ | $28.2$ | $25.6$ |

Ex2 (open) | $21.7$ | $16.8$ | $19.2$ | $19.2$ |

Ex3 (open) | $18.3$ | $16.3$ | $16.9$ | $17.2$ |

Ex4 (apt.) | $57.9$ | $54.9$ | $49.3$ | $54.0$ |

Ex5 (apt.) | $42.8$ | $41.5$ | $39.7$ | $41.3$ |

Ex6 (apt.) | $39.2$ | $42.7$ | $39.0$ | $40.3$ |

$\widehat{\mathit{\mu}}$ [dB] | $\widehat{\mathit{\varphi}}$ [dB] | $\widehat{\mathit{\lambda}}$ [m] | ${\widehat{\mathit{\sigma}}}^{2}$ [dB${}^{2}$] | |
---|---|---|---|---|

EKF | $0.1705$ | $1.3688$ | $0.0433$ | $0.6324$ |

ARTI | $0.6623$ | $2.5230$ | $0.0365$ | $4.1551$ |

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## Share and Cite

**MDPI and ACS Style**

Kaltiokallio, O.; Hostettler, R.; Yiğitler, H.; Valkama, M.
Unsupervised Learning in RSS-Based DFLT Using an EM Algorithm. *Sensors* **2021**, *21*, 5549.
https://doi.org/10.3390/s21165549

**AMA Style**

Kaltiokallio O, Hostettler R, Yiğitler H, Valkama M.
Unsupervised Learning in RSS-Based DFLT Using an EM Algorithm. *Sensors*. 2021; 21(16):5549.
https://doi.org/10.3390/s21165549

**Chicago/Turabian Style**

Kaltiokallio, Ossi, Roland Hostettler, Hüseyin Yiğitler, and Mikko Valkama.
2021. "Unsupervised Learning in RSS-Based DFLT Using an EM Algorithm" *Sensors* 21, no. 16: 5549.
https://doi.org/10.3390/s21165549